Is the US ready for a third president called Bush?

In the firmament of the Bushes, it was meant to be George W's week, what with the dedication of his presidential library and an attendant spate of TV interviews, not to mention the rare gathering of all five living US presidents and ex-presidents to celebrate the occasion. In an odd way, though, last week belonged to Jeb.

The term "library" is a bit of a misnomer. These peculiarly American institutions, covering every presidency since Herbert Hoover's, are indeed repositories of official papers, and managed by the federal government at taxpayers' expense. But their construction is privately financed, and for the average visitor they are primarily museums, designed – not surprisingly – to present the men who raised the money for them in the best possible light.

If you doubt me, try Richard Nixon's at his birthplace in Yorba Linda, California. True, the section dealing with a small matter called Watergate has been considerably beefed up, but overall the place retains an almost reverential air. And if the rehabilitation of the much-reviled "43" is to start anywhere, it will be at the George W Bush Presidential Library and Museum on the campus of Southern Methodist University in Dallas.

On Thursday, even Barack Obama got into the act, describing Bush as "a good man" and praising his "incredible strength and resolve" in the immediate aftermath of 9/11. Not a word about Iraq, and not for an instant would you have imagined that he won an election six months ago by blaming everything wrong in the country on his predecessor.

But right now, even on Dubya's big day in Dallas, the word association of "Bush" and "president" denotes not only him, or even his father, George H W Bush. Increasingly, the calculation involves Jeb, George's youngest brother, the former two-term governor of Florida – and probably, at this absurdly early moment in the ante-post betting, the bookies' favourite for the Republican presidential nomination in 2016.

America is supposed to be the land of opportunity for all. Increasingly, though, its politics can bring to mind Kirchners, Aliyevs and Assads. Once upon a time, US dynasties referred to financiers, industrialists and sports teams. Now, almost as much as in Argentina, Azerbaijan or Syria, they mean presidents. Between 1976 and 2012, not an election went by without featuring either a Bush or a Clinton in the primaries, and in 1992 the families squared off for the White House. What odds against a repeat?

A year ago, the prospect looked unlikely. Hillary Clinton was a visibly exhausted Secretary of State, telling one and all that she intended to retire after Obama's first term and give herself over to writing and teaching. As for Jeb, "this was probably my time", he mused wistfully in June 2012, soon after Mitt Romney, had clinched the nomination in one of the weakest Republican fields in memory.

But, of course, Jeb couldn't have run then. The Bush brand was anathema; to have offered the country the chance of choosing another one so soon would have been, if not hubris, then a guarantee that a reporter's first question would be, what was your brother's biggest mistake?

Since then everything has changed. Romney was soundly beaten by Obama, and these days it's not so much the Bush brand, but the entire Republican/Tea Party brand, well to the right of even George W, that's in the doghouse. Jeb, a fiscal and social conservative but a keen proponent of immigration reform, with a Mexican wife and perfect Spanish, now looks the answer to his party's problems with Hispanic voters, whose desertion to Obama sealed Romney's defeat. As for Hillary, polls showing 70-plus per cent support are a wonderful antidote to weariness.

And there's a further symmetry to a Bush/Clinton rematch. Even in her student days, her peers predicted that if America were ever to have a female president, it would be Hillary. As for Jeb, the smoothest, brainiest and most accomplished of George H W and Barbara Bush's children, he was always seen by the father as bearer of the family's political torch.

All that shifted, however, on 8 November 1994, when Jeb, the favourite, narrowly failed to be elected governor of Florida, while George Jnr, the ne'er-do-well, pulled off an upset victory over Texas's popular incumbent, Ann Richards. Six years later, with Karl Rove still at his side, Jeb's brother won the White House.

Now, maybe, Jeb's time has truly come. But on one condition – that America, unlike four years ago, is ready once more for a Bush. The problem, of course, is in the name. As Haley Barbour, the ex-governor of Mississippi and wisest of old Republican birds, recently put it: "If Jeb's last name was Brown instead of Bush, he'd probably be the front-runner for the Republican nomination. Then again, if it were Brown, we probably would have never heard of him."

Jeb himself does nothing to discourage speculation. He gives weighty speeches about rebuilding America. He's published a book on immigration reform, and is even allowing dirty linen (a former family maid who was deported for being an illegal immigrant) to be washed in public now, before opponents wash it in the heat of a campaign. A final decision, he says, will come after the 2014 mid-terms. As with Hillary, name recognition allows him to wait – and a Clinton run could silence complaints that Jeb is running on his name. Dynasties? Everyone does dynasties.

Except, perhaps, some complaints from within the family. Not from George W, who when asked last week for his advice to Jeb about 2016, replied, "Run". But Barbara, at 88 more than ever the outspoken matriarch, feels differently. "It's a great country," she says, "it's not just four families or whatever. We've had enough Bushes." But, this time, the son may not listen to his mother. And if he runs and wins, a presidential library awaits.